(i) What is binding energy of a nucleus ? Write its expression. (ii) Compute the binding energy of ⁴₂He nucleu

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Q. (i) What is binding energy of a nucleus ? Write its expression.

(ii) Compute the binding energy of

^{4}_{2}\text{He}

nucleus using the following data : Atomic mass of Helium atom,

M_A(\text{He}) = 4.00260

u, Mass of hydrogen atom,

m_H = 1.00785

u, and Mass of neutron,

m_n = 1.008665

u. OR

(i) Write about impedance of the series RLC circuit.

(ii) Find the impedance of a series RLC circuit if the inductive reactance, capacitive reactance and resistance are 184

\Omega

, 144

\Omega

and 30

\Omega

respectively. Also calculate the phase angle between voltage and current.

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Concept understanding — Nuclear Binding Energy per Nucleon

The Intuition: Why Are Nuclei Stuck Together?

Imagine a nucleus as a tight cluster of protons and neutrons. Protons all carry positive charge, so they should be violently repelling each other. Yet the nucleus holds together. That means there must be an even stronger attractive force — the strong nuclear force — acting between nucleons (protons and neutrons). But here's the catch: this force is extremely short-range. A nucleon only feels the pull from its immediate neighbours, not from nucleons far across the nucleus.

So the nucleus is a tug-of-war. The strong force pulls nucleons together, but the electrostatic repulsion between protons tries to blow the nucleus apart. For a nucleus to be stable, the strong force must win.

Now, if you want to break a nucleus apart into its individual protons and neutrons, you have to do work against the strong force — you have to supply energy. That energy, once supplied, is stored in the separated nucleons as extra mass. This is the core idea: the mass of a stable nucleus is always less than the sum of the masses of its individual protons and neutrons. The missing mass is called the mass defect, and the energy equivalent of that missing mass (via $E = mc^2$ ) is the binding energy.

Note

Binding energy is the energy you must put in to completely separate a nucleus into its constituent nucleons. It is not energy stored inside the nucleus like fuel; it is the energy that holds the nucleus together.

The Precise Statement

For a nucleus with $Z$ protons, $N$ neutrons, and mass $m_{\text{nucleus}}$ :

\text{Mass defect } \Delta m = Z m_p + N m_n - m_{\text{nucleus}}

where $m_p$ and $m_n$ are the masses of a free proton and neutron. Then the total binding energy is:

\text{BE} = \Delta m \, c^2

But a big nucleus has more nucleons, so its total binding energy will naturally be larger. To compare how tightly bound different nuclei are, we use binding energy per nucleon:

\text{BE/A} = \frac{\text{BE}}{A}

where $A = Z + N$ is the mass number. This is the average energy you'd need to remove one nucleon from the nucleus. A higher BE/A means a more stable nucleus.

The Famous Curve: Why Iron is Special

If you plot BE/A against mass number $A$ , you get a curve that rises steeply for light nuclei, peaks at iron-56 (about 8.8 MeV per nucleon), and then slowly falls for heavier nuclei.

Important

Iron-56 has the highest binding energy per nucleon of any nuclide. It is the most stable nucleus in nature.

This shape tells you two things:

1. Fusion of light nuclei releases energy. If you take two very light nuclei (like hydrogen isotopes) and smash them together to form a medium-mass nucleus, the product has a higher BE/A than the reactants. The difference in binding energy is released as kinetic energy of the products. This is how stars burn.

2. Fission of heavy nuclei releases energy. If you split a very heavy nucleus (like uranium-235) into two medium-mass fragments, those fragments also have higher BE/A than the original. Again, the difference comes out as energy. This is how nuclear reactors work. …

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(i) Binding energy is the mass-defect energy $\Delta m\,c^2$ holding a nucleus together; for He-4 it works out to about 28.35 MeV. (ii)/OR: a series RLC circuit's impedance combines resistance and net reactance as $Z=\sqrt{R^2+(X_L-X_C)^2}$ , giving Z=50 Ω and phase angle ≈53.1° for the given values. Both alternatives answered below.

(i) Binding energy of a nucleus -- definition

The mass of any stable nucleus is less than the sum of the masses of its free, separated constituent nucleons -- this difference is the mass defect $\Delta m$ . By Einstein's mass-energy equivalence, this missing mass corresponds to an energy

$BE = \Delta m\,c^2$

called the binding energy of the nucleus -- physically, the minimum energy that would need to be supplied to completely separate the nucleus into its individual free protons and neutrons (equivalently, the energy released when the nucleus was originally assembled from free nucleons). Using atomic mass units and the conversion $1\,\text{u}=931.5$ MeV:

$BE\,(\text{MeV}) = \Delta m\,(\text{u}) \times 931.5$

(ii) Binding energy of $^4_2\text{He}$

Mass defect (using atomic masses, so the atomic electrons cancel consistently on both sides):

$\Delta m = \big[2\,m_H + 2\,m_n\big] - M_A(\text{He})$

$= \big[2(1.00785) + 2(1.008665)\big] - 4.00260$

$= [2.01570 + 2.01733] - 4.00260 = 4.03303 - 4.00260 = 0.03043\ \text{u}$

Binding energy:

$BE = 0.03043 \times 931.5 \approx 28.35\ \text{MeV}$

(Binding energy per nucleon $\approx 7.09$ MeV, one of the highest among light nuclei, reflecting helium-4's unusual stability.)

OR (i) Impedance of a series RLC circuit -- definition

In a series RLC circuit, the resistor's voltage is in phase with the current, while the inductor's and capacitor's voltages are $90°$ out of phase with the current (in opposite senses). Combining these by phasor (vector) addition, the total opposition to current -- the impedance $Z$ -- is

$Z = \sqrt{R^2+(X_L-X_C)^2}$ …

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Q.(i) What is binding energy of a nucleus ? Write its expression.

(i) Binding energy of a nucleus -- definition

(ii) Binding energy of 24He^4_2\text{He}24​He

OR (i) Impedance of a series RLC circuit -- definition

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(ii) Binding energy of $^4_2\text{He}$

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